I have the same problem with my Moto E4 running Android 7.1.1. What’s different in my case is that the blink app used to work just fine, ever since I got the phone in 2019. Then, just a few weeks ago, it would no longer stay signed in between uses. It started requiring me to re-signin sometimes, and then every time, even if I hadn’t closed the app. Now I cannot sign in at all. It does exactly what you’ve reported, refreshing the signin screen with no error message and no option to do anything else.
This sure feels like the problem is with the app, not the phone. The phone hasn’t changed, but Blink has updated its app multiple times in recent weeks. I’m working now by email with a Blink support person who is very responsive and articulate. He checked their system, and found it showed that my phone was actually signed in. He cleared that status, and I tried again (uninstall, reboot, reinstall, try to sign in) but I’m still getting the same problem. I’ll report further as the troubleshooting process progresses.